System and method for designing of dictionaries for sparse representation

ABSTRACT

A signal processing system adapted for sparse representation of signals is provided, comprising: (i) one or more training signals; (ii) a dictionary containing signal-atoms; (iii) a representation of each training signal using a linear combination of said dictionary&#39;s signal-atoms; (iv) means for updating the representation of the training signal; (v) means for updating the dictionary one group of atoms at a time, wherein each atom update may include all representations referring to said updated atom; and (vi) means for iterating (iv) and (v) until a stopping rule is fulfilled. The system uses the K-SVD algorithm for designing dictionaries for sparse representation of signals.

FIELD OF THE INVENTION

The present invention relates to a system and method for the sparserepresentation of signals. The invention is particularly relevant forapplications such as compression, regularization in inverse problems,feature extraction, denoising, separation of texture and cartoon contentin images, signal analysis, signal synthesis, inpainting andrestoration.

BACKGROUND OF THE INVENTION

Sparse Representation of Signals

In recent years there has been a growing interest in the study of sparserepresentation of signals. Using an overcomplete dictionary thatcontains prototype signal-atoms, signals are described as sparse linearcombinations of these atoms. Applications that use sparse representationare many and include compression, regularization in inverse problems,feature extraction, and more. Recent activity in this field concentratedmainly on the study of pursuit algorithms that decompose signals withrespect to a given dictionary. Designing dictionaries to better fit theabove model can be done by either selecting one from a pre-specified setof linear transforms, or by adapting the dictionary to a set of trainingsignals. Both these techniques have been considered in recent years, butthis topic is largely still open.

Using an overcomplete dictionary matrix Dε

^(n×K) that contains K prototype signal-atoms for columns, {d_(j)}_(j=1)^(K), it is assumed that a signal yε

^(n) can be represented as a sparse linear combination of these atoms.The representation of y may either be exact y=Dx, or approximate, y≈Dx,satisfying |y−Dx′|_(p)≦ε. The vector xε

^(K) displays the representation coefficients of the signal y. Inapproximation methods, typical norms used for measuring the deviationare the l^(p)-norms for p=1, 2 and ∞.

If n<K and D is a full-rank matrix, an infinite number of solutions areavailable for the representation problem, hence constraints on thesolution must be set. The solution with the fewest number of nonzerocoefficients is certainly an appealing representation. This, sparsestrepresentation, is the solution of either

$\begin{matrix}{{{\left( P_{0} \right)\;{\min\limits_{x}{{x}_{0}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu} y}}} = {Dx}},{or}} & (1) \\{{{\left( P_{0,\varepsilon} \right){\min\limits_{x}{{x}_{0}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{{y - {Dx}}}_{2}}}} \leq \varepsilon},} & (2)\end{matrix}$where ∥•∥₀ is the l⁰ norm, counting the non zero entries of a vector.

Applications that can benefit from the sparsity and overcompletenessconcepts (together or separately) include compression, regularization ininverse problems, feature extraction, and more. Indeed, the success ofthe JPEG2000 coding standard can be attributed to the sparsity of thewavelet coefficients of natural images. In denoising (removal of noisefrom noisy data so to obtain the unknown or original signal), waveletmethods and shift-invariant variations that exploit overcompleterepresentation, are among the most effective known algorithms for thistask. Sparsity and overcompleteness have been successfully used fordynamic range compression in images, separation of texture and cartooncontent in images, inpainting (changing an image so that the change isnot noticeable by an observer) and restoration, and more.

In order to use overcomplete and sparse representations in applications,one needs to fix a dictionary D, and then find efficient ways to solve(1) or (2). Recent activity in this field has been concentrated mostlyon the study of so called pursuit algorithms that represent signals withrespect to a known dictionary, and approximate the solutions of (1) and(2). Exact determination of sparsest representations proves to be anNP-hard problem. Thus, approximate solutions are considered instead, andseveral efficient pursuit algorithms have been proposed in the lastdecade. The simplest ones are the Matching Pursuit (MP) and theOrthogonal Matching Pursuit (OMP) algorithms. Both are greedy algorithmsthat select the dictionary atoms sequentially. These methods are verysimple, involving the computation of inner products between the signaland dictionary columns' and possibly deploying some least squaressolvers. Both (1) and (2) are easily addressed by changing the stoppingrule of the algorithm.

A second well known pursuit approach is the Basis Pursuit (BP). Itsuggests a convexisation of the problems posed in (1) and (2), byreplacing the l⁰-norm with an l¹-norm. The Focal Under-determined SystemSolver (FOCUSS) is very similar, using the l^(p)-norm with p≦1, as areplacement to the l⁰-norm. Here, for p<1 the similarity to the truesparsity measure is better, but the overall problem becomes non-convex,giving rise to local minima that may divert the optimization. Lagrangemultipliers are used to convert the constraint into a penalty term, andan iterative method is derived based on the idea of iterated reweighedleast-squares that handles the l^(p)-norm as an l² weighted one.

Both the BP and FOCUSS can be motivated based on Maximum A Posteriori(MAP) estimation and indeed several works used this reasoning directly.The MAP can be used to estimate the coefficients as random variables bymaximizing the posterior P(x|y,D)αP(y|D,x)P(x). The prior distributionon the coefficient vector x is assumed to be a super-GaussianIndependent Identically-Distributed (iid) distribution that favorssparsity. For the Laplace distribution this approach is equivalent toBP.

Extensive study of these algorithms in recent years has established thatif the sought solution, x, is sparse enough, these techniques recover itwell in the exact case. Further work considered the approximatedversions and has shown stability in recovery of x. The recent front ofactivity revisits those questions within a probabilistic setting,obtaining more realistic assessments on pursuit algorithms performanceand success. The properties of the dictionary D set the limits that maybe assumed on the sparsity that consequently ensure successfulapproximation. Interestingly, in all the works mentioned so far, thereis a preliminary assumption that the dictionary is known and fixed.There is a great need to address the issue of designing the properdictionary in order to better fit the sparsity model imposed.

The Choice of the Dictionary

An overcomplete dictionary D that leads to sparse representations caneither be chosen as a pre-specified set of functions, or designed byadapting its content to fit a given set of signal examples.

Choosing a pre-specified transform matrix is appealing because it issimpler. Also, in many cases it leads to simple and fast algorithms forthe evaluation of the sparse representation. This is indeed the case forovercomplete wavelets, curvelets, contourlets, steerable waveletfilters, short-time-Fourier transforms, and more. Preference istypically given to tight frames that can easily be pseudo-inverted. Thesuccess of such dictionaries in applications depends on how suitablethey are to sparsely describe the signals in question. Multiscaleanalysis with oriented basis functions and a shift-invariant propertyare guidelines in such constructions.

There is need to develop a different route for designing dictionaries Dbased on learning, and find the dictionary D that yields sparserepresentations for the training signals. Such dictionaries have thepotential to outperform commonly used pre-determined dictionaries. Withever-growing computational capabilities, computational cost may becomesecondary in importance to the improved performance achievable bymethods which adapt dictionaries for special classes of signals.

Sparse coding is the process of computing the representationcoefficients, x, based on the given signal y and the dictionary D. Thisprocess, commonly referred to as “atom decomposition”, requires solving(1) or (2), and this is typically done by a “pursuit algorithm” thatfinds an approximate solution. Three popular pursuit algorithms are theOrthogonal Matching Pursuit (OMP), Basis Pursuit (BP) and the FocalUnder-determined System Solver (FOCUSS).

Orthogonal Matching Pursuit is a greedy step-wise regression algorithm.At each stage this method selects the dictionary element having themaximal projection onto the residual signal. After each selection, therepresentation coefficients with regarding to the so far chosen atomsare found via least-squares. Formally, given a signal yε

^(n), and a dictionary D with K l²-normalized columns {d_(k)}_(k=1)^(K), one starts by setting r₀=y, k=1, and performing the followingsteps:

1) Select the index of the next dictionary elementi_(k)=argmax_(w)|<r_(k−1),d_(w)>|;

2) Update the current approximation y_(k)=argmin_(y) _(k) ∥y−y_(k)∥₂ ²such that y_(k)εspan{d_(i) ₁ , d_(i) ₂ , . . . , d_(i) _(k) }; and

3) Update the residual r_(k)=y−y_(k).

The algorithm can be stopped after a predetermined number of steps,hence after having selected a fixed number of atoms. Alternatively, thestopping rule can be based on norm of the residual, or on the maximalinner product computed in the next atom selection stage.

OMP is an appealing and very simple to implement algorithm. Unlike othermethods, it can be easily programmed to supply a representation with ana priori fixed number of non-zero entries—a desired outcome in thetraining of dictionaries. There are several variants of the OMP thatsuggest (i) skipping the least-squares and using inner product itself asa coefficient, or (ii) applying least-squares per every candidate atom,rather than just using inner-products at the selection stage, or (iii)doing faster and less precise search, where instead of searching for themaximal inner product, a nearly maximal one is selected, therebyspeeding up the search.

Theoretic study has shown that the OMP solution is also the sparsestavailable one (solving (1)) if some conditions on the dictionary and onthe exact solution prevail. More recent work has shown that the above isalso true for the approximation version (2). These results and somelater ones that apply to the basis pursuit and FOCUSS involve a keyfeature of the dictionary D called the mutual incoherence and definedas:

$\begin{matrix}{\mu = {\max\limits_{i \neq j}{{{d_{i}^{T}d_{j}}}.}}} & (3)\end{matrix}$

This measure quantifies how similar two columns of the dictionary canbe. Given μ, if the sparse representation to be found has fewer than

(1/μ) non-zeros, the OMP and its variants are guaranteed to succeed inrecovering it.

Basis Pursuit (BP) algorithm proposes the replacement of the l⁰-norm in(1) and (2) with an l¹-norm. Hence solutions of:

$\begin{matrix}{{{\left( P_{1} \right)\;{\min\limits_{x}{{x}_{1}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu} y}}} = {Dx}},} & (4)\end{matrix}$in the exact representation case, and

$\begin{matrix}{{{\left( P_{1,\varepsilon} \right){\min\limits_{x}{{x}_{1}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{{y - {Dx}}}}}} \leq \varepsilon},} & (5)\end{matrix}$in the approximate one, lead to the BP representations. Solution of (4)amounts to linear programming, and thus there exists efficient solversfor such problems.

Recent research addressed the connection between the (P₀) and (P₁). Theessential claims are quite similar to the ones of OMP, namely, if thesignal representation to be found has fewer than

(1/μ) non-zeros, the BP is guaranteed to succeed in recovering it.Similar results exist for the approximated case, proving that recoveredrepresentations are very close to the original sparse one in case ofhigh sparsity.

Focal Under-determined System Solver (FOCUSS) is an approximatingalgorithm for finding the solutions of either (1) or (2), by replacingthe l⁰-norm with an l^(p) one for p≦1.

For the exact case problem, (P₀), this method requires solving

$\begin{matrix}{{\left( P_{p} \right)\;{\min\limits_{x}{{x}_{p}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu} y}}} = {{Dx}.}} & (6)\end{matrix}$

The use of a Lagrange multiplier vector λε

here yields the Lagrangian function

(x,λ)=∥x∥ _(p)+λ^(T)(y−Dx).  (7)

Hence necessary conditions for a pair x, λ to be a solution of 6 are

_(x)

(x,λ)=pπ(x)x−D ^(T)λ=0 and

_(λ)

(x,λ)=Dx−y=0,  (8)where π(x) is defined to be a diagonal matrix with |x_(i)|^(p−2)| as its(i, i)^(th) entry. The split of the l^(p)-norm derivative into a linearterm multiplied by a weight matrix is the core of the FOCUSS method, andthis follows the well-known idea of iterated reweighed least-squares.Several simple steps of algebra leads to the solution:x=π(x)⁻¹ D ^(T)(Dπ(x)⁻¹ D ^(T))⁻¹ y.  (9)

While it is impossible to get a closed form solution for x from theabove result, an iterative replacement procedure can be proposed, wherethe right hand side is computed based on the currently known x_(k−1),and this leads to the updating process,x _(k)=π(x _(k−1))⁻¹ D ^(T)(Dπ(x _(k−1))⁻¹ D ^(T))⁻¹ y.  (10)

A regularization can, and should, be introduced to avoid near-zeroentries in the weight matrix π(x).

For the treatment of (P_(0,ε)) via the (P_(p,ε)) parallel expressionscan be derived quite similarly, although in this case the determinationof the Lagrange multiplier is more difficult and must be searched withinthe algorithm.

Recent work analyzed the (P_(p)) problem and showed its equivalence tothe (P₀), under conditions similar in flavor to the sparsity conditionsmentioned above. Hence, this approach too enjoys the support of sometheoretical justification, like BP and OMP. However, the analysis doesnot say anything about local minima traps and prospects in hitting thosein the FOCUSS-algorithm.

Design of Dictionaries: Prior Art

There has been some work in the field regarding the training ofdictionaries based on a set of examples. Given such set Y={y_(i)}_(i=1)^(N), we assume that there exists a dictionary D that gave rise to thegiven signal examples via sparse combinations, i.e., we assume thatthere exists D, so that solving (P₀) for each example y_(k) gives asparse representation x_(k). It is in this setting that the question israised what the proper dictionary D is.

A. Generalizing the K-Means

There is an intriguing relation between sparse representation andclustering (i.e., vector quantization). In clustering, a set ofdescriptive vectors {d_(k)}_(k=1) ^(K) is learned, and each sample isrepresented by one of those vectors (the one closest to it, usually inthe l² distance measure). One can think of this as an extreme sparserepresentation, where only one atom is allowed in the signaldecomposition, and furthermore, the coefficient multiplying it mustbe 1. There is a variant of the vector quantization (VQ) coding method,called Gain-Shape VQ, where this coefficient is allowed to vary. Incontrast, in sparse representations relevant to the invention, eachexample is represented as a linear combination of several vectors{d_(k)}_(k=1) ^(K). Thus, sparse representations can be referred to as ageneralization of the clustering problem.

Since the K-Means algorithm (also known as generalized Lloydalgorithm—GLA) is the most commonly used procedure for training in thevector quantization setting, it is natural to consider generalizationsof this algorithm when turning to the problem of dictionary training.The K-Means process applies two steps per each iteration: (i) given{d_(k)}_(k=1) ^(K), assign the training examples to their nearestneighbor; and (ii) given that assignment, update {d_(k)}_(k=1) ^(K) tobetter fit the examples.

The approaches to dictionary design that have been tried so far are verymuch in line with the two-step process described above. The first stepfinds the coefficients given the dictionary—a step we shall refer to as“sparse coding”. Then, the dictionary is updated assuming known andfixed coefficients. The differences between the various algorithms thathave been proposed are in the method used for the calculation ofcoefficients, and in the procedure used for modifying the dictionary.

B. Maximum Likelihood Methods

Maximum likelihood methods use probabilistic reasoning in theconstruction of D. The proposed model suggests that for every example ythe relationy=Dx+v,  (11)holds true with a sparse representation x and Gaussian white residualvector v with variance σ². Given the examples Y={y_(i)}_(i=1) ^(N) theseworks consider the likelihood function P (Y|D) and seek the dictionarythat maximizes it. Two assumptions are required in order to proceed—thefirst is that the measurements are drawn independently, readilyproviding

$\begin{matrix}{{P\left( Y \middle| D \right)} = {\prod\limits_{i = 1}^{N}{{P\left( y_{i} \middle| D \right)}.}}} & (12)\end{matrix}$

The second assumption is critical and refers to the “hidden variable” x.The ingredients of the likelihood function are computed using therelationP(y _(i) |D)=∫P(y _(i) ,x|D)dx=∫P(y _(i) |x·D)·P(x)dx.  (13)

Returning to the initial assumption in (11), we have

$\begin{matrix}{{P\left( {\left. y_{i} \middle| x \right.,D} \right)} = {{{Const} \cdot \exp}{\left\{ {\frac{1}{2\sigma^{2}}{{{Dx} - y_{i}}}^{2}} \right\}.}}} & (14)\end{matrix}$

The prior distribution of the representation vector x is assumed to besuch that the entries of x are zero-mean iid, with Cauchy or Laplacedistributions. Assuming for example a Laplace distribution we get

$\begin{matrix}\begin{matrix}{{P\left( y_{i} \middle| D \right)} = {\int{{{P\left( {\left. y_{i} \middle| x \right.,D} \right)} \cdot {P(x)}}{\mathbb{d}x}}}} \\{= {{Const} \cdot {\int{\exp{\left\{ {\frac{1}{2\sigma^{2}}{{{Dx} - y_{i}}}^{2}} \right\} \cdot \exp}\left\{ {\lambda{x}_{1}} \right\}{\mathbb{d}x}}}}}\end{matrix} & (15)\end{matrix}$

This integration over x is difficult to evaluate, and indeed, it hasbeen handled by replacing it with the extremal value of P(y_(i),x|D).The overall problem turns into

$\begin{matrix}\begin{matrix}{D = {\arg\;{\max\limits_{D}{\sum\limits_{i = 1}^{N}{\max\limits_{x_{i}}\left\{ {P\left( {y_{i},\left. x_{i} \middle| D \right.} \right)} \right\}}}}}} \\{= {\arg\;{\min\limits_{D}{\sum\limits_{i = 1}^{N}{\min\limits_{x_{i}}\left\{ {{{{Dx}_{i} - y_{i}}}^{2} + {\lambda{x_{i}}_{1}}} \right\}}}}}}\end{matrix} & (16)\end{matrix}$

This problem does not penalize the entries of D as it does for the onesof x_(i). Thus, the solution will tend to increase the dictionaryentries' values, in order to allow the coefficients to become closer tozero. This difficulty has been handled by constraining the l²-norm ofeach basis element, so that the output variance of the coefficients iskept at an appropriate level.

An iterative method was suggested for solving (16). It includes two mainsteps in each iteration: (i) calculate the coefficients x_(i) using asimple gradient descent procedure; and then (ii) update the dictionaryusing

$\begin{matrix}{D^{({n + 1})} = {D^{(n)} - {\eta{\sum\limits_{i = 1}^{N}\;{\left( {{D^{(n)}x_{i}} - y_{i}} \right)x_{i}^{T}}}}}} & (17)\end{matrix}$

This idea of iterative refinement, mentioned before as a generalizationof the K-Means algorithm, was later used again by other researchers,with some variations.

A different approach to handle the integration in (15) has beensuggested. It consisted in approximating the posterior as a Gaussian,enabling an analytic solution of the integration. This allows anobjective comparison of different image models (basis or priors). Italso removes the need for the additional re-scaling that enforces thenorm constraint. However, this model may be too limited in describingthe true behaviors expected. This technique and closely related oneshave been referred to as approximated ML techniques.

There is an interesting relation between the maximum likelihood methodand the Independent Component Analysis (ICA) algorithm. The latterhandles the case of a complete dictionary (the number of elements equalsthe dimensionality) without assuming additive noise. The maximumlikelihood method is then similar to ICA in that the algorithm can beinterpreted as trying to maximize the mutual information between theinputs (samples) and the outputs (the coefficients).

C. The Method of Optimal Directions

The Method of Optimal Directions (MOD), a dictionary-training algorithm,follows more closely the K-Means outline, with a sparse coding stagethat uses either the OMP or FOCUSS, followed by an update of thedictionary. The main contribution of the MOD method is its simple way ofupdating the dictionary.

Assuming that the sparse coding for each example is known, we define theerrors e_(i)=y_(i)−Dx_(i). The overall representation mean square erroris given by∥E∥ _(F) ² =∥[e ₁ ,e ₂ , . . . ,e _(N)]∥_(F) ² =∥Y−DX∥ _(F) ².  (18)Here we have concatenated all the examples y_(i) as columns of thematrix Y, and similarly gathered the representations coefficient vectorsx_(i) to build the matrix X. The notation ∥A∥_(F) stands for theFrobenius Norm, defined as ∥A∥_(F)=√{square root over (Σ_(ij)A_(ij) ²)}.

Assuming that X is fixed, we can seek an update to D such that the aboveerror is minimized. Taking the derivative of (10) with respect to D weobtain the relation (Y−DX)X^(T)=0, leading toD ^((n+1)) =YX ^((n)) ^(T) ·(X ^((n)) X ^((n)) ^(T) )⁻¹  (19)

In updating the dictionary, the update relation given in (19) is thebest that can be achieved for fixed X. The iterative steepest descentupdate in (17) is far slower. Interestingly, in both stages of thealgorithm, the difference is in deploying a second order (Newtonian)update instead of a first-order one. Looking closely at the updaterelation in (17), it could be written as

$\begin{matrix}\begin{matrix}{D^{({n + 1})} = {D^{(n)} + {\eta\;{EX}^{{(n)}^{T}}}}} \\{= {D^{(n)} + {{\eta\left( {Y - {D^{(n)}X^{(n)}}} \right)}X^{{(n)}^{T}}}}} \\{= {{D^{(n)}\left( {I - {\eta\; X^{(n)}X^{{(n)}^{T}}}} \right)} + {\eta\;{{YX}^{{(n)}^{T}}.}}}}\end{matrix} & (20)\end{matrix}$

Using infinitely many iterations of this sort, and using small enough η,this leads to a steady state outcome, which is exactly the MOD updatematrix (19). Thus, while the MOD method assumes known coefficients ateach iteration, and derives the best possible dictionary, the ML methodby Olshausen and Field only gets closer to this best current solution,and then turns to calculate the coefficients. Note, however, that inboth methods a normalization of the dictionary columns is required anddone.

D. Maximum A-Posteriori Probability Approach

The same researchers that conceived the MOD method also suggested amaximum a-posteriori probability (MAP) setting for the training ofdictionaries, attempting to merge the efficiency of the MOD with anatural way to take into account preferences in the recovereddictionary. This probabilistic point of view is very similar to the MLmethods discussed above. However, rather than working with thelikelihood function P(Y|D), the posterior P(D|Y) is used. Using Bayesrule, we have P(D|Y)αP(Y|D)P(D), and thus we can use the likelihoodexpression as before, and add a prior on the dictionary as a newingredient.

Research currents considered several priors P(D) and per each proposedan update formula for the dictionary. The efficiency of the MOD in thesemethods is manifested in the efficient sparse coding, which is carriedout with FOCUSS. The proposed algorithms in this family deliberatelyavoid a direct minimization with respect to D as in MOD, due to theprohibitive n×n matrix inversion required. Instead, iterative gradientdescent is used.

When no prior is chosen, the update formula is the very one used in(17). A prior that constrains D to have a unit Frobenius norm leads tothe update formulaD ^((n+1)) −D ^((n)) +ηEX ^(T) +η·tr(XE ^(T) D ^((n)))D ^((n)).  (21)

As can be seen, the first two terms are the same ones as in (17). Thelast term compensates for deviations from the constraint. This caseallows different columns in D to have different norm values. As aconsequence, columns with small norm values tend to be under-used, asthe coefficients they need are larger and as such more penalized.

This led to the second prior choice, constraining the columns of D tohave a unit l²-norm. The new update equation formed is given byd _(i) ^((n+1)) =d _(i) ^((n))+η(I−d _(i) ^((n)) d _(i) ^((n)) ^(T) )E·_(i) ^(T),  (22)where x^(T) _(i) is the i-th column in the matrix X^(T).

Compared to the MOD, this line of work provides slower trainingalgorithms.

E. Unions of Orthogonal Bases

Recent work considered a dictionary composed as a union of orthonormalbasesD=[D₁;D₂, . . . ,D_(L)],where D_(j)ε

^(n×n), j=1, 2, . . . , L are orthonormal matrices. Such a dictionarystructure is quite restrictive, but its updating may potentially be mademore efficient.

The coefficients of the sparse representations X can be decomposed to Lpieces, each referring to a different ortho-basis. Thus,X=[X₁,X₂, . . . ,X_(L)]^(T),where X_(i) is the matrix containing the coefficients of the orthonormaldictionary Di.

One of the major advantages of the union of ortho-bases is the relativesimplicity of the pursuit algorithm needed for the sparse coding stage.The coefficients are found using the Block Coordinate Relaxation (BCR)algorithm. This is an appealing way to solve (P_(1,ε)) as a sequence ofsimple shrinkage steps, such that at each stage X_(i) is computed, whilekeeping all the other pieces of X fixed. Thus, this evaluation amountsto a simple shrinkage.

Assuming known coefficients, the proposed algorithm updates eachorthonormal basis D_(j) sequentially. The update of D_(j) is done byfirst computing the residual matrix

$E_{j} = {\left\lbrack {e_{1},e_{2},\ldots\mspace{14mu},e_{N}} \right\rbrack = {Y - {\sum\limits_{i \neq j}^{\;}\;{D_{i}{X_{i}.}}}}}$

Then, by computing the singular value decomposition of the matrixE_(j)X^(T) _(j)=UΛV^(T), the update of the j-th ortho-basis is done byD_(j)=UV^(T). This update rule is obtained by solving a constrainedleast squares problem with ∥E_(j)−D_(j)X_(j)∥_(F) ² as the penalty term,assuming fixed coefficients X_(j) and error E_(j). The constraint isover the feasible matrices D_(j), which are forced to be orthonormal.

This way the proposed algorithm improves each matrix D_(j) separately,by replacing the role of the data matrix Y in the residual matrix E_(j),as the latter should be represented by this updated basis.

Grinbonval suggested a slightly different method. Apart from the factthat here the dictionary is structured, handling a union of orthonormalbases, it updates each orthonormal bases sequentially, and thus remindsthe sequential updates done in the K-means. Experimental results showweak performance compared to previous methods. This could partly beexplained by the fact that the update of D_(j) depends strongly on thecoefficients X_(j).

K-Means Algorithm for Vector Quantization (VQ)

In VQ, a codebook C that includes K codewords (representatives) is usedto represent a wide family of vectors (signals) Y={y_(i)}_(i=1) ^(N) (N

K) by a nearest neighbor assignment. This leads to an efficientcompression or description of those signals, as clusters in

surrounding the chosen codewords. Based on the expectation maximizationprocedure, the K-Means can be extended to suggest a fuzzy assignment anda covariance matrix per each cluster, so that the data is modeled as amixture of Gaussians.

The dictionary of VQ codewords is typically trained using the K-Meansalgorithm. We denote the codebook matrix by C=[c₁, c₂, . . . , c_(K)],the codewords being the columns. When C is given, each signal isrepresented as its closest codeword (under l²-norm distance). We canwrite y_(i)=Cx_(i), where x_(i)=e_(j) is a vector from the trivialbasis, with all zero entries except a one in the j-th position. Theindex j is selected such that∀_(k≠j) ∥y _(i) −Ce _(j)∥₂ ² ≦∥y _(i) −Ce _(k)∥₂ ².

This is considered as an extreme case of sparse coding in the sense thatonly one atom is allowed to participate in the construction of y_(i),and the coefficient is forced to be 1. The representation MSE per y_(i)is defined ase _(i) ² =∥y _(i) −Cx _(i)∥₂ ².  (23)and the overall MSE is

$\begin{matrix}{E = {{\sum\limits_{i = 1}^{K}e_{i}^{2}} = {{{Y - {CX}}}_{F}^{2}.}}} & (24)\end{matrix}$The VQ training problem is to find a codebook C that minimizes the errorE, subject to the limited structure of X, whose columns must be takenfrom the trivial basis,

$\begin{matrix}{{\min\limits_{C,X}{\left\{ {{Y - {CX}}}_{F}^{2} \right\}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{\forall i}}},{x_{i\;} = {e_{k}\mspace{14mu}{for}\mspace{14mu}{some}\mspace{14mu}{k.}}}} & (25)\end{matrix}$

The K-Means algorithm is an iterative method used for designing theoptimal codebook for VQ. In each iteration there are two stages—one forsparse coding that essentially evaluates X, and one for updating thecodebook.

The sparse coding stage assumes a known codebook C^((J−1)), and computesa feasible X that minimizes the value of (25). Similarly, the dictionaryupdate stage fixes X as known, and seeks an update of C so as tominimize (25). Clearly, at each iteration either a reduction or nochange in the MSE is ensured. Furthermore, at each such stage, theminimization step is optimal under the assumptions. As the MSE isbounded from below by zero, and the algorithm ensures a monotonicdecrease of the MSE, convergence to at least a local minimum solution isguaranteed. Stopping rules for the above-described algorithm can vary alot but are quite easy to handle.

SUMMARY OF THE PRIOR ART

Almost all previous methods can essentially be interpreted asgeneralizations of the K-Means algorithm, and yet, there are markeddifferences between these procedures. In the quest for a successfuldictionary training algorithm, there are several desirable properties:

(i) Flexibility: the algorithm should be able to run with any pursuitalgorithm, and this way enable choosing the one adequate for therun-time constraints, or the one planned for future usage in conjunctionwith the obtained dictionary. Methods that decouple the sparse-codingstage from the dictionary update readily have such a property. Such isthe case with the MOD and the MAP based methods.

(ii) Simplicity: much of the appeal of a proposed dictionary trainingmethod has to do with how simple it is, and more specifically, howsimilar it is to K-Means. It is desirable to have an algorithm that maybe regarded as a natural generalization of the K-Means. The algorithmshould emulate the ease with which the K-Means is explainable andimplementable. Again, the MOD seems to have made a substantial progressin this direction, although there is still room for improvement.

(iii) Efficiency: the proposed algorithm should be numerically efficientand exhibit fast convergence. The above described methods are all quiteslow. The MOD, which has a second-order update formula, is nearlyimpractical in reasonable dimensions, because of the matrix inversionstep involved. Also, in all the above formulations, the dictionarycolumns are updated before turning to re-evaluate the coefficients. Thisapproach inflicts a severe limitation on the training speed.

(iv) Well Defined Objective: for a method to succeed, it should have awell defined objective function that measures the quality of thesolution obtained. This almost trivial fact was overlooked in some ofthe preceding work in this field. Hence, even though an algorithm can bedesigned to greedily improve the representation Mean Square Error (MSE)and the sparsity, it may happen that the algorithm leads to aimlessoscillations in terms of a global objective measure of quality.

SUMMARY OF THE INVENTION

It is an object of the present invention to design a dictionary based onlearning from training signals, wherein the dictionary yields sparserepresentations for a set of training signals. These dictionaries havethe potential to outperform commonly used pre-determined dictionaries.

The invention thus relate to a novel system and algorithm for adaptingdictionaries so as to represent signals sparsely. Given a set oftraining signals {y_(i)}_(i=1) ^(N), we seek the dictionary D that leadsto the best possible representations for each member in this set withstrict sparsity constraints. The invention introduces the K-SVDalgorithm that addresses the above task, generalizing the K-Meansalgorithm. The K-SVD is an iterative method that alternates betweensparse coding of the examples based on the current dictionary, and anupdate process for the dictionary atoms so as to better fit the data.The update of the dictionary columns is done jointly with an update ofthe sparse representation coefficients related to it, resulting inaccelerated convergence. The K-SVD algorithm is flexible and can workwith any pursuit method, thereby tailoring the dictionary to theapplication in mind.

The sparse representation problem can be viewed as a generalization ofthe VQ objective (25), in which we allow each input signal to berepresented by a linear combination of codewords, which we now calldictionary elements. Therefore the coefficients vector is now allowedmore than one nonzero entry, and these can have arbitrary values. Forthis case, the minimization corresponding to Equation (25) is that ofsearching the best possible dictionary for the sparse representation ofthe example set Y.

$\begin{matrix}{{\min\limits_{D,X}{\left\{ {{Y - {DX}}}_{F}^{2} \right\}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{\forall i}}},{{x_{i}}_{0} \leq {T_{0}.}}} & (26)\end{matrix}$

A similar objective could alternatively be met by considering

$\begin{matrix}{{{{\min\limits_{D,X}{\sum\limits_{i}{{x_{i}}_{0}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{{Y - {DX}}}_{F}^{2}}}} \leq} \in},} & (27)\end{matrix}$for a fixed value ε. By disclosing the treatment for the first problem(26), any person skilled in the art would immediately realize that thetreatment is very similar.

In the algorithm of the invention, we minimize the expression in (26)iteratively. First, we fix D and aim to find the best coefficient matrixX that can be found. As finding the truly optimal X is impossible, weuse an approximation pursuit method. Any such algorithm can be used forthe calculation of the coefficients, as long as it can supply a solutionwith a fixed and predetermined number of nonzero entries, T₀.

Once the sparse coding task is done, a second stage is performed tosearch for a better dictionary. This process updates one column at atime, fixing all columns in D except one, d_(k), and finding a newcolumn d_(k) and new values for its coefficients that best reduce theMSE. This is markedly different from all the K-Means generalizations.K-Means generalization methods freeze X while finding a better D. Theapproach of the invention is different, as we change the columns of Dsequentially, and allow changing the relevant coefficients. In a sense,this approach is a more direct generalization of the K-Means algorithm,because it updates each column separately, as done in K-Means.

The process of updating only one column of D at a time is a problemhaving a straightforward solution based on the singular valuedecomposition (SVD). Furthermore, allowing a change in the coefficients'values while updating the dictionary columns accelerates convergence,since the subsequent columns updates will be based on more relevantcoefficients. The overall effect is very much in line with the leap fromgradient descent to Gauss-Seidel methods in optimization.

A hypothetical alternative would be to skip the step of sparse coding,and use only updates of columns in D, along with their coefficients,applied in a cyclic fashion, again and again. This however will not workwell, as the support of the representations will never be changed, andsuch an algorithm will necessarily fall into a local minimum trap.

The invention is useful for a variety of applications in signalprocessing including but not limited to: compression, regularization ininverse problems, feature extraction, denoising, separation of textureand cartoon content in images, signal analysis, signal synthesis,inpainting and restoration.

Typically, all the training signals involved are from the same familyand thus have common traits. For examples, the signals can all bepictures, music, speech etc.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a description of the K-SVD algorithm of the invention.

FIG. 2 is graph of synthetic results comparing K-SVD against two knownalgorithms, MOD and MAP-based algorithms. For each of the testedalgorithms and for each noise level, 50 trials were conducted and theirresults sorted. The graph labels represent the mean number of detectedatoms (out of 50) over the ordered tests in groups of 10 experiments.

FIG. 3 is a collection of 500 random block patches of size 8×8 pixels,taken from a database of face images, which were used for training withthe K-SVD algorithm, sorted by their variance.

FIG. 4A depicts the learned dictionary (a K-SVD trained dictionary ofsize 64×441). Its elements are sorted in an ascending order of theirvariance, and stretched to maximal range for display purposes. FIGS. 4Band 4C depict the overcomplete separable Haar dictionary and theovercomplete DCT dictionary, respectively, of the same size (shown forcomparison).

FIG. 5 is the RMSE for 594 new blocks with missing pixels using thelearned dictionary of FIG. 4A, the overcomplete Haar dictionary and theovercomplete DCT dictionary.

FIGS. 6A-6H compare two corrupted images and their reconstruction, withthe missing pixels marked as points (6A, 50% of missing pixels; 6E, 70%of missing pixels), and the reconstructed results by the learneddictionary (6B, 6F), the overcomplete Haar dictionary (6C, 6G), and theovercomplete DCT dictionary (6D, 6H), respectively.

FIG. 7 depicts Rate-Distortion graphs for the compression results foreach dictionary.

FIGS. 8A-8C show sample compression results for the K-SVD, overcompleteDCT and complete DCT dictionaries, respectively.

DETAILED DESCRIPTION OF THE INVENTION

In the following detailed description of various embodiments, referenceis made to the accompanying drawings that form a part thereof, and inwhich are shown by way of illustration specific embodiments in which theinvention may be practiced. It is understood that other embodiments maybe utilized and structural changes may be made without departing fromthe scope of the present invention.

In the present invention, we address the problem of designingdictionaries, and introduce the K-SVD algorithm for this task. We showhow this algorithm can be interpreted as a generalization of the K-Meansclustering process, and demonstrate its behavior in both synthetic testsand in applications on real data.

The present invention relates to a signal processing method adapted forsparse representation of signals and a system for implementing saidmethod, said system comprising:

-   -   (i) one or more training signals;    -   (ii) a dictionary containing signal-atoms;    -   (iii) a representation of each training signal using a linear        combination of said dictionary's signal-atoms;    -   (iv) means for updating the representation of the training        signal;    -   (v) means for updating the dictionary one group of atoms at a        time, wherein each atom update may include all representations        referring to said updated atom; and    -   (vi) means for iterating (iv) and (v) until a stopping rule is        fulfilled.

The training signals are typically from the same family and thus alltraining signals share common traits and have common behavior patterns.For example, all training signals can be pictures, including pictures ofhuman faces, or the training signals can be sound files including musicfiles, speeches, and the like.

The purpose of the dictionary of the present invention is to discoverthe common building blocks with which all the training signals can berepresented. All the training signals can be represented by linearcombinations of the dictionary atoms (building blocks). The term “atom”as referred to herein means dictionary atom or signal-atom.

In some cases, the building blocks or some of the building blocks of thetraining signals are known or can be approximated intuitively, while inother cases the invention helps to discover them.

According to a preferred embodiment, the dictionary is updated one atomat a time. It is possible however to also update the dictionary a groupof atoms at a time, for example two or three atoms at a time, ordefining the group of atoms to be updated containing any number ofatoms.

In one embodiment of the present invention, the dictionary is anovercomplete dictionary. An overcomplete dictionary contains more atoms(building blocks, functions) than strictly necessary to represent thesignal space. An overcomplete dictionary thus allows a suitablerepresentation of a signal with fewer encoded atoms. This is importantfor applications in which a low bit rate is required.

Signals can be represented in many forms. In one embodiment of thepresent invention, the representation of each training signal is acoefficient matrix. The representation of the training signals may takeany other form such as a vector.

There are many ways to generate a coefficient matrix representing thetraining signals. In one embodiment of the present invention, thegeneration of the coefficients matrix is achieved by a pursuitalgorithm. The pursuit algorithm can include: Orthogonal MatchingPursuit, Matching Pursuit, Basis Pursuit, FOCUSS or any combination orvariation thereof.

Updating the dictionary can be performed sequentially or in any otherorder. In yet another embodiment of the present invention, thedictionary is updated in a predefined order of the signal-atoms.Depending on the application used and the nature of the trainingsignals, updating the dictionary in a predefined order of signal-atomswill yield different results and thus can be exploited by theapplication.

In another embodiment of the present invention, only selectedsignal-atoms of said dictionary are updated. Again, depending on thenature of the application in mind, one may decide to leave certainsignal-atoms (building blocks) fixed, and consequently only update theremaining signal-atoms.

In some cases, it may happen that a dictionary is built wherein twosignal-atoms are very similar to each other but not equal to each other.The similarity, for the purpose of the application used, may be too big,and thus the differentiation between the two atoms may be considerednegligible. The application will thus wish to modify one of the similaratoms. In yet another embodiment of the present invention, a signal-atomis modified when the difference between said signal-atom to anothersignal atom is below a predefined value.

A signal-atom may be defined by the system as a building block forrepresenting the training signals, but the actual signal-atom may neverbe used to construct any of the given training signals. One may thuswish to modify this atom. In a further embodiment of the presentinvention, a signal-atom is modified when it is not used in anyrepresentation.

A signal-atom may be found to be used only rarely to construct trainingsignals. It may be thus preferred not to work with such a buildingblock, and modify this atom to one used more frequently in trainingsignals representation. In one embodiment of the present invention, asignal-atom is modified when its usage frequency in the representationof signal-atoms is below a predefined value.

When updating the dictionary, either a single atom or a group of atomsat a time, there are many possibilities to define the best results forthe atom values. In yet another embodiment of the present invention,updating the group of atoms and their coefficients best reduces the MeanSquare Error (MSE).

In some cases, again depending on the nature of the application used andof the training signals, it may be desired to design dictionaries withone or more custom properties. For example, the dictionary can beshift-invariant. A system is shift-invariant if f(x−α,y−β)→g(x−α,y−β)for arbitrary α and β. Another embodiment of the invention may design adictionary with non-negative dictionary values, wherein each atomcontains only non-negative entries. Another option is to force zeros inpredetermined places in the dictionary. It is possible to design thedictionary with any matrix structure. Multiscale dictionaries or zerosin predefined places are two examples of a structure, but any structurecan be used depending on the nature of the training signals andapplication in mind. A person skilled in the art will easily designother properties in the dictionary according the training signals andthe nature of the application. Such custom properties are all consideredto be with the scope of the present invention.

In yet another embodiment of the present invention, multiscaledictionaries are built. An image, for example, can be defined usingmultiscale dictionaries, wherein each dictionary represents the image ina different size. Obviously, a smaller image will show fewer detailsthan a bigger image.

The invention can be used for a variety of applications, including butnot limited to: for compression, regularization in inverse problems,feature extraction, denoising, separation of texture and cartoon contentin images, signal analysis, signal synthesis, inpainting andrestoration.

The K-SVD—Detailed Description

As mentioned previously, the objective function of the K-SVD is

$\begin{matrix}{{\min\limits_{D,X}{\left\{ {{Y - {DX}}}_{F}^{2} \right\}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{\forall i}}},{{x_{i}}_{0} \leq {T_{0}.}}} & (28)\end{matrix}$

Let us first consider the sparse coding stage, where we assume that D isfixed, and consider the optimization problem as a search for sparserepresentations with coefficients summarized in the matrix X. Thepenalty term can be rewritten as

${{Y - {DX}}}_{F}^{2} = {\sum\limits_{i = 1}^{N}{{{y_{i} - {Dx}_{i}}}_{2}^{2}.}}$Therefore the problem posed in (28) can be decoupled to N distinctproblems of the form

$\begin{matrix}{{i = 1},{2\ldots}\mspace{14mu},N,{{\min\limits_{x_{i}}{\left\{ {{y_{i} - {Dx}_{i}}}_{2}^{2} \right\}\mspace{14mu}{subject}\mspace{14mu}{to}\mspace{14mu}{x_{i}}_{0}}} \leq {T_{0}.}}} & (29)\end{matrix}$

This problem is adequately addressed by the pursuit algorithms mentionedbefore, and we have seen that if T₀ is small enough, their solution is agood approximation to the ideal one that is numerically infeasible tocompute.

We now turn to the second, and slightly more involved process ofupdating the dictionary together with the nonzero coefficients. Assumethat both X and D are fixed, and we put in question only one column inthe dictionary, d_(k), and the coefficients that correspond to it, thei-th row in X, denoted as x^(i) _(T) (this is not the vector xi which isthe i-th column in X). Returning to the objective function (28), thepenalty term can be rewritten as

$\begin{matrix}\begin{matrix}{{{Y - {DX}}}_{F}^{2} = {{Y - {\sum\limits_{j = 1}^{K}{d_{j}x_{T}^{j}}}}}_{F}^{2}} \\{= {{\left( {Y - {\sum\limits_{j \neq k}^{\;}{d_{j}x_{T}^{j}}}} \right) - {d_{k}x_{T}^{k}}}}_{F}^{2}} \\{= {{{E_{k} - {d_{k}x_{T}^{k}}}}_{F}^{2}.}}\end{matrix} & (30)\end{matrix}$

We have decomposed the multiplication DX to the sum of K rank-1matrices. Among those, K−1 terms are assumed fixed, and one—thek-th—remains in question. The matrix E_(k) stands for the error for allthe N examples when the k-th atom is removed.

Here, it would be tempting to suggest the use of the SVD (Singular ValueDecomposition) to find alternative d_(k) and x^(k) _(T). The SVD findsthe closest rank-1 matrix (in Frobenius norm) that approximates E_(k),and this will effectively minimize the error as defined in (30).However, such a step will be a mistake, because the new vector x^(k)_(T) is very likely to be filled, since in such an update of d_(k) we donot enforce the sparsity constraint.

A remedy to the above problem, however, is simple and also quiteintuitive. Define w_(i) as the group of indices pointing to examples{y_(i)} that use the atom d_(k), i.e., those where x^(k) _(T)(i) isnonzero. Thus,ω_(k) ={i|1≦i≦K·x _(T) ^(k)(i)≠0}.  (31)

Define Ω_(k) as a matrix of size N×|w_(i)|, with ones on the (w_(k)(i),i)-th entries, and zeros elsewhere. When multiplying x^(k) _(R)=x^(k)_(T)Ωk, this shrinks the row vector x^(k) _(T) by discarding of the zeroentries, resulting with the row vector x^(k) _(R) of length |w_(k)|.Similarly, the multiplication Y^(R) _(k)=YΩ_(k) creates a matrix of sizen×|w_(k)| that includes a subset of the examples that are currentlyusing the d_(k) atom. The same effect happens with E^(R)_(k)=E_(k)Ω_(k), implying a selection of error columns that correspondto examples that use the atom d_(k).

With this notation, we can now return to (30) and suggest minimizationwith respect to both d_(k) and x^(k) _(T), but this time force thesolution of x^(k) _(T) to have the same support as the original x^(k)_(T). This is equivalent to the minimization of∥E _(k)Ω_(k) −d _(k) x _(T) ^(k)Ω_(k)∥_(F) ² =∥E _(k) ^(R) −d _(k) x_(R) ^(k)∥_(F) ²,  (32)and this time it can be done directly via SVD. Taking the restrictedmatrix E^(R) _(k), SVD decomposes it to E^(R) _(k)=UΔV^(T). We definethe solution for d_(k) as the first column of U, and the coefficientvector x^(k) _(R) as the first column of V multiplied by Δ(1, 1). Inthis solution, we necessarily have that (i) the columns of D remainnormalized; and (ii) the support of all representations either stays thesame or gets smaller by possible nulling of terms.

This algorithm has been herein named “K-SVD” to parallel the nameK-Means. While K-Means applies K computations of means to update thecodebook, the K-SVD obtains the updated dictionary by K SVDcomputations, each determining one column. A full description of thealgorithm is given in FIG. 1.

In the K-SVD algorithm we sweep through the columns and use always themost updated coefficients as they emerge from preceding SVD steps.Parallel versions of this algorithm can also be considered, where allupdates of the previous dictionary are done based on the same X.Experiments show that while this version also converges, it yields aninferior solution, and typically requires more than 4 times the numberof iterations. These parallel versions and variation are all encompassedby the present invention.

An important question that arises is: Will the K-SVD algorithm converge?Let us first assume we can perform the sparse coding stage perfectly,retrieving the best approximation to the signal y_(i) that contains nomore than T₀ nonzero entries. In this case, and assuming a fixeddictionary D, each sparse coding step decreases the total representationerror ∥Y−DX∥² _(F), posed in (28). Moreover, at the update step ford_(k), an additional reduction or no change in the MSE is guaranteed,while not violating the sparsity constraint. Executing a series of suchsteps ensures a monotonic MSE reduction, and therefore, convergence to alocal minimum is guaranteed.

Unfortunately, the above statement depends on the success of pursuitalgorithms to robustly approximate the solution to (29), and thusconvergence is not always guaranteed. However, when T₀ is small enoughrelative to n, the OMP, FOCUSS, and BP approximating methods are knownto perform very well. While OMP can be naturally used to get a fixed andpre-determined number of non-zeros (T₀), both BP and FOCUSS require someslight modifications. For example, in using FOCUSS to lead to T₀non-zeros, the regularization parameter should be adapted whileiterating. In those circumstances the convergence is guaranteed. We canensure convergence by external interference—by comparing the bestsolution using the already given support to the one proposed by the newrun of the pursuit algorithm, and adopting the better one. This way weshall always get an improvement. Practically, we saw in all ourexperiments that a convergence is reached, and there was no need forsuch external interference.

From K-SVD Back to K-Means

When the model order T₀=1, this case corresponds to the gain-shape VQ,and as such it is important, as the K-SVD becomes a method for itscodebook training. When T₀=1, the coefficient matrix X has only onenonzero entry per column. Thus, computing the error E^(R) _(k) in (32),yields

$\begin{matrix}{E_{k}^{R} = {{E_{k}\Omega_{k}} = {{\left( {Y - {\sum\limits_{j \neq k}^{\;}{d_{j}x_{T}^{j}}}} \right)\Omega_{k}} = {{Y\;\Omega_{k}} = {Y_{k}^{R}.}}}}} & (33)\end{matrix}$This is because the restriction Ω_(k) takes only those columns in E_(k)that use the d_(k) atom, and thus, necessarily, they use no other atoms,implying that for all j, x_(T) ^(j)Ω_(k)=0.

The implication of the above outcome is that the SVD in the T₀=1 case isdone directly on the group of examples in w_(k). Also, the K updates ofthe columns of D become independent of each other, implying that asequential process as before, or a parallel one, both lead to the samealgorithm.

We could further constraint our representation stage and beyond thechoice T₀=1, limit the nonzero entries of X to be 1. This brings us backto the classical clustering problem as described earlier. In this casewe have that x^(k) _(R) is filled with ones, thus x^(k) _(R)=1^(T). TheK-SVD then needs to approximate the restricted error matrix E^(R)_(k)=Y^(R) _(k) by a rank-1 matrix d_(k)·1^(T). The solution is the meanof the columns of Y^(R) _(k), exactly as K-Means suggests.

K-SVD—Implementation Details

Just like the K-Means, the K-SVD algorithm is susceptible to localminimum traps. Our experiments show that improved results can be reachedif the following variations are applied:

(i) When using approximation methods with a fixed number ofcoefficients, we found out that FOCUSS proves to be the best in terms ofgetting the best out of each iteration. However, from a run-time pointof view, OMP was found to lead to far more efficient overall algorithm.

(ii) When a dictionary element is not being used “enough” (relative tothe number of dictionary elements and to the number of samples) it couldbe replaced with the least represented data element, after beingnormalized (the representation is measured without the dictionaryelement that is going to be replaced). Since the number of data elementsis much larger than the number of dictionary elements, and since ourmodel assumption suggests that the dictionary atoms are of equalimportance, such replacement is very effective in avoiding local minimaand over-fitting.

(iii) Similar to the idea of removal of unpopular elements from thedictionary, we found that it is very effective to prune the dictionaryfrom having too-close elements. If indeed such a pair of atoms is found(based on their absolute inner product exceeding some threshold), one ofthose elements should be removed and replaced with the least-representedsignal.

Similarly to the K-Means, we can propose a variety of techniques tofurther improve the K-SVD algorithm. Appealing examples on this list aremulti-scale approaches and tree-based training where the number ofcolumns K is allowed to increase during the algorithm. All thesevariations, adaptations and improvements are encompassed by the presentinvention.

Synthetic Experiments

We have first tried the K-SVD algorithm on synthetic signals, to testwhether this algorithm recovers the original dictionary that generatedthe data, and to compare its results with other reported algorithms.

Step 1—generation of the data to train on: A random matrix D (referredto later-on as the generating dictionary) of size 20×50 was generatedwith iid uniformly distributed entries. Each column was normalized to aunit l²-norm. Then, 1500 data signals {y_(i)}_(i=1) ¹⁵⁰⁰ of dimension 20were produced, each created by a linear combination of 3 differentgenerating dictionary atoms, with uniformly distributed iid coefficientsin random and independent locations. White Gaussian noise with varyingSignal to Noise Ration (SNR) was added to the resulting data signals.

Step 2—applying the K-SVD: The dictionary was initialized with datasignals. The coefficients were found using OMP with fixed number of 3coefficients. The maximum number of iterations was set to 80.

Step 3—comparison to other reported works: we implemented the MODalgorithm, and applied it on the same data, using OMP with fixed numberof 3 coefficients, and initializing in the same way. We executed the MODalgorithm for a total number of 80 iterations. We also executed theMAP-based algorithm of Rao and Kreutz-Delgado (Kreutz-Delgado et al.,Dictionary learning algorithms for sparse representation. NeuralComputation. 15(2):349-396, 2003). This algorithm was executed as is,therefore using FOCUSS as its decomposition method. Here, again, amaximum of 80 iterations were allowed.

Results: the computed dictionary was compared against the knowngenerating dictionary. This comparison was done by sweeping through thecolumns of the generating dictionary, and finding the closest column (inl² distance) in the computed dictionary, measuring the distance via1−|d_(i) ^(T){tilde over (d)}_(i)|,  (34)where d_(i) is a generating dictionary atom, and {tilde over (d)}_(i) isits corresponding element in the recovered dictionary, A distance lessthan 0.01 was considered a success. All trials were repeated 50 times,and the number of successes in each trial was computed. The results forthe three algorithms and for noise levels of 10 dB, 20 dB, 30 dB and ∞dB (no noise) are displayed in FIG. 2.

We should note that for different dictionary size (e.g., 20×30) and withmore executed iterations, the MAP-based algorithm improves and getscloser to the K-SVD detection rates.

Applications to Image Processing

Several experiments have been conducted on natural image data, trying toshow the practicality of the algorithm of the invention and the generalsparse coding theme. These preliminary tests prove the concept of usingsuch dictionaries with sparse representations.

Training Data: The training data was constructed as a set of 11,000examples of block patches of size 8×8 pixels, taken from a database offace images (in various locations). A random collection of 500 suchblocks, sorted by their variance, is presented in FIG. 3.

Removal of the DC: Working with real images data we preferred that alldictionary elements except one has a zero mean. For this purpose, thefirst dictionary element, denoted as the DC, was set to include aconstant value in all its entries, and was not changed afterwards. TheDC takes part in all representations, and as a result, all otherdictionary elements remain with zero mean during all iterations.

Running the K-SVD: We applied the K-SVD, training a dictionary of size64×441. The choice K=441 came from our attempt to compare the outcome tothe overcomplete Haar dictionary of the same size. The coefficients werecomputed using the OMP with fixed number of coefficients, where themaximal number of coefficients is 10, A better performance can beobtained by switching to FOCUSS. The test was conducted using OMPbecause of its simplicity and fast execution. The trained dictionary ispresented in FIG. 4A.

Comparison Dictionaries: The trained dictionary was compared with theovercomplete Haar dictionary which includes separable basis functions,having steps of various sizes and in all locations (total of 441elements). In addition, we built an overcomplete separable version ofthe DCT dictionary by sampling the cosine wave in different frequenciesto result a total of 441 elements. The overcomplete Haar dictionary ispresented in FIG. 4B and the overcomplete DCT dictionary is presented inFIG. 4C.

Applications. The K-SVD results were used, denoted here as the learneddictionary, for two different applications on images. All tests wereperformed on one face image which was not included in the training set.The first application is filling-in missing pixels: random pixels in theimage were deleted, and their values were filled using the variousdictionaries decomposition. Then the compression potential of thelearned dictionary decomposition was tested, and a rate-distortion graphwas presented. These experiments will be described in more detailhereafter.

A. Filling-In Missing Pixels

One random full face image was chosen, which consists of 594non-overlapping blocks (none of which were used for training). For eachblock, the following procedure was conducted for r in the range {0.2,0.9}:

(i) A fraction r of the pixels in each block, in random locations, weredeleted (set to zero).

(ii) The coefficients of the corrupted block under the learneddictionary, the overcomplete Haar dictionary, and the overcomplete DCTdictionary were found using OMP with an error bound of ∥0.02·1∥₂, where1εR^(n) is a vector of all ones (the input image is scald to the dynamicrange [0, 1]), (allowing an error of ±5 gray-values in 8-bit images).All projections in the OMP algorithm included only the non-corruptedpixels, and for this purpose, the dictionary elements were normalized sothat the non-corrupted indices in each dictionary element have a unitnorm. The resulting coefficient vector of the block B is denoted x_(B).

(iii) The reconstructed block {tilde over (B)} was chosen as {tilde over(B)}=D·x_(B).

(iv) The reconstruction error was set to: √{square root over (∥B−{tildeover (B)}∥_(F) ²/64)} (64 is the number of pixels in each block). Themean reconstruction errors (for all blocks and all corruption rates)were computed, and are displayed in FIG. 5. Two corrupted images andtheir reconstructions can be seen in FIGS. 6A-6H. FIG. 6A shows a facewith 50% missing pixels, where FIGS. 6B, 6C and 6D show a learnedreconstruction, a Haar reconstruction and a complete DCT reconstructionrespectively. FIG. 6E shows a face with 70% missing pixels, where FIGS.6F, 6G and 6H show a learned reconstruction, a Haar reconstruction and acomplete DCT reconstruction respectively. As can be seen, higher qualityrecovery is obtained using the learned dictionary.

B. Compression

A compression comparison was conducted between the overcomplete learneddictionary, the overcomplete Haar dictionary, and the overcomplete DCTdictionary (as described before), all of size 64×441. In addition, acomparison was made to the regular (unitary) DCT dictionary (used by theJPEG algorithm). The resulted rate-distortion graph is presented in FIG.7. In this compression test, the face image was partitioned (again) into594 disjoint 8×8 blocks. All blocks were coded in various rates(bits-per-pixel values), and the Peak Signal-to-Noise Ratio (PSNR) wasmeasured. Let I be the original image and Ĩ be the coded image, combinedby all the coded blocks. We denote ε² as the mean squared error betweenI and Ĩ, and calculate

$\begin{matrix}{{PSNR} = {10 \cdot {{\log_{10}\left( \frac{1}{e^{2}} \right)}.}}} & (35)\end{matrix}$

In each test we set an error goal ε, and fixed the number ofbits-per-coefficient Q. For each such pair of parameters, all blockswere coded in order to achieve the desired error goal, and thecoefficients were quantized to the desired number of bits (uniformquantization, using upper and lower bounds for each coefficient in eachdictionary based on the training set coefficients). For the overcompletedictionaries, the OMP coding method was used. The rate value was definedas

$\begin{matrix}{{R = \frac{{{a \cdot \#}\mspace{11mu}{Blocks}} + {\#\;{{coefs} \cdot \left( {b + Q} \right)}}}{\#\mspace{11mu}{pixels}}},} & (36)\end{matrix}$where

-   -   a holds the required number of bits to code the number of        coefficients for each block.    -   b holds the required number of bits to code the index of the        representing atom. Both a and b values were calculated using an        entropy coder.    -   #Blocks is the number of blocks in the image (594).    -   #coefs is the total number of coefficients required to represent        the whole image.    -   #pixels is the number of pixels in the image (=64×#Blocks).        In the unitary DCT dictionary the coefficients were picked in a        zigzag order, as done by JPEG, until the error bound ε is        reached. Therefore, the index of each atom should not be coded,        and the rate was defined by,

$\begin{matrix}{{R = \frac{{{a \cdot \#}\mspace{11mu}{Blocks}} + {\#\;{{coefs} \cdot Q}}}{\#\mspace{11mu}{pixels}}},} & (37)\end{matrix}$with the same notation as before.

By sweeping through various values of ε and Q we get per each dictionaryseveral curves in the R-D plane. FIG. 7 presents the best obtained R-Dcurves for each dictionary. As can be seen, the K-SVD dictionaryoutperforms all other dictionaries, and achieves up to 1-2 dB better forbit rates less than 1.5 bits-per-pixel (where the sparsity model holdstrue). Samples results are presented in FIGS. 8A-8C. FIG. 8A shows theresult using the K-SVD dictionary, while FIGS. 8B and 8C show theresults using the overcomplete DCT dictionary and the complete DCTdictionary, respectively.

Although the invention has been described in detail, neverthelesschanges and modifications, which do not depart from the teachings of thepresent invention, will be evident to those skilled in the art. Suchchanges and modifications are deemed to come within the purview of thepresent invention and the appended claims.

1. A computerized signal processing method configured for sparserepresentation of signals, said method comprising the steps of: (i)reading one or more training signals; (ii) defining a dictionarycontaining signal-atoms; (iii) creating a representation of eachtraining signal using a linear combination of said dictionary'ssignal-atoms; (iv) updating the representation of the training signal;(v) updating the dictionary a single atom or one group of atoms at atime, wherein each atom update includes an update of all representationsreferring to said updated atom; and (vi) iterating (iv) and (v) until astopping rule is fulfilled.
 2. The computerized signal processing methodaccording to claim 1, wherein said group of atoms contains one atomonly.
 3. The computerized signal processing method according to claim 1,wherein all training signals belong to the same family and have one ormore common traits.
 4. The computerized signal processing methodaccording to claim 1, wherein said dictionary is an overcompletedictionary.
 5. The computerized signal processing method according toclaim 1, wherein the representation of all training signals is acoefficient matrix.
 6. The computerized signal processing methodaccording to claim 1, wherein the generation of each coefficients matrixis achieved by a pursuit algorithm.
 7. The computerized signalprocessing method according to claim 6, wherein said pursuit algorithmincludes Orthogonal Matching Pursuit, Matching Pursuit, Basis Pursuit,FOCUSS or any combination or variation thereof.
 8. The computerizedsignal processing method according to claim 1, wherein said dictionaryis updated in a predefined order of the signal-atoms.
 9. Thecomputerized signal processing method according to claim 1, wherein onlyselected signal-atoms of said dictionary are updated.
 10. Thecomputerized signal processing method according to claim 1, wherein asignal-atom is modified when it is not used in any representation. 11.The computerized signal processing method according to claim 10, whereina signal-atom is modified when the difference between said signal-atomto another signal atom is below a predefined value.
 12. The computerizedsignal processing method according to claim 1, wherein a signal-atom ismodified when its usage frequency in the representation of signal-atomsis below a predefined value.
 13. The computerized signal processingmethod according to claim 1, wherein updating said group of atoms andtheir coefficients best reduces the Mean Square Error.
 14. Thecomputerized signal processing method according to claim 1, wherein saiddictionary is designed with custom properties.
 15. The computerizedsignal processing method according to claim 14, wherein said customproperties include: (vii) shift invariant; (viii) non-negativedictionary values so that each atom contains only non-negative entries;(ix) zeros in predetermined places; and (x) any structure of a matrix.16. The computerized signal processing method according to claim 1,comprising multiscale dictionaries.
 17. The computerized signalprocessing method according to claim 1, for compression, regularizationin inverse problems, feature extraction, denoising, separation oftexture and cartoon content in images, signal analysis, signalsynthesis, inpainting and restoration.